〔一〕今有句三尺，股四尺，问为弦几何？
荅曰：五尺。
现在有一个直角三角形，其中一条直角边长为三尺，另一条直角边长为四尺，问斜边的长度是多少？
答：五尺。
Now there is a right triangle, one of whose legs measures three feet, and the other leg measures four feet. What is the length of the hypotenuse? Answer: Five feet.
〔二〕今有弦五尺，句三尺，问为股几何？
荅曰：四尺。
现在有一根弦长五尺，弧长三尺，问股长多少？
答案是四尺。
Now there is a chord of five feet, and an arc of three feet; what is the length of the leg? The answer is four feet.
〔三〕今有股四尺，弦五尺，问为句几何？
荅曰：三尺。句股术曰：句股各自乘，并，而开方除之，即弦。
又股自乘，以减弦自乘，其余开方除之，即句。
又句自乘，以减弦自乘，其余开方除之，即股。
现在有一个直角三角形，其中一条直角边长为四尺，斜边长为五尺，问另一条直角边的长度是多少？
答：三尺。《九章算术》中说：将直角边自乘后相加，再开方，即可得到斜边的长度。
将一条直角边自乘后，用斜边自乘的数减去它，再开方，即可得到另一条直角边的长度。
将另一条直角边自乘后，用斜边自乘的数减去它，再开方，即可得到第一条直角边的长度。
Now, there is a right triangle with one leg measuring four feet and the hypotenuse measuring five feet. How long is the other leg? Answer: Three feet. The "Nine Chapters on the Mathematical Art" states: Square each leg of the right triangle and add them, then take the square root to find the length of the hypotenuse. Square one leg and subtract it from the square of the hypotenuse, then take the square root to find the length of the other leg. Square the other leg and subtract it from the square of the hypotenuse, then take the square root to find the length of the first leg.
〔四〕今有圆材径二尺五寸，欲为方版，令厚七寸。问广几何？
荅曰：二尺四寸。
术曰：令径二尺五寸自乘，以七寸自乘减之，其余开方除之，即广。
现有直径为二尺五寸的圆形木材，想要制作成厚度为七寸的方形板块。问这个方板的宽度是多少？
答案是二尺四寸。方法是：将直径二尺五寸自乘，再以七寸自乘减去它，剩下的开方后除以它，就得到了宽度。
We currently have a round timber with a diameter of two feet and five inches. We wish to make it into a square board with a thickness of seven inches. What is the width? The answer is: two feet and four inches. The method is: square the diameter of two feet and five inches, subtract the square of seven inches from it, take the square root of the remainder, and divide by it to get the width.
〔五〕今有木长二丈，围之三尺。葛生其下，缠木七周，上与木齐。问葛长几何？
荅曰：二丈九尺。
术曰：以七周乘三尺为股，木长为句，为之求弦。弦者，葛之长。
现在有一棵树高两丈，树围三尺。葛藤生长在树下，缠绕树木七圈，顶端与树齐平。问葛藤有多长？
回答是：两丈九尺。
解法是：将七圈乘以三尺作为股，树的高度作为句，求出弦的长度。这个弦，就是葛藤的长度。
Now there is a tree that is two zhang in height, with a circumference of three chi. A kudzu vine grows beneath it, winding around the tree for seven revolutions, with its top level with the tree. How long is the kudzu? The answer is: two zhang and nine chi. The method is: multiply seven revolutions by three chi to get the length of the leg, and use the height of the tree as the hypotenuse to find the length of the string. This string, is the length of the kudzu.
〔六〕今有池方一丈，葭生其中央，出水一尺。引葭赴岸，適与岸齐。问水深、葭长各几何？。
荅曰：
水深一丈二尺；葭长一丈三尺。
术曰：半池方自乘，以出水一尺自乘，减之，余，倍出水除之，即得水深。加出水数，得葭长
现有一个池塘，边长为一丈，芦苇生长在池塘中央，露出水面一尺。将芦苇拉向岸边，恰好与岸边齐平。问水深和芦苇的长度各是多少？
答案是：
水深一丈二尺；芦苇长一丈三尺。
方法是：取半个池塘边长的平方，减去露出水面一尺的平方，剩下的再翻倍后除以2，就得到水深。然后加上露出水面的一尺，就得到芦苇的长度。
We have a square pond with sides that are 10 feet long. A reed grows in the middle of the pond, protruding one foot above the water. When the reed is pulled to the bank, it aligns exactly with the edge of the pond. What is the depth of the water and the length of the reed? The answer is: The depth of the water is 12 feet; the length of the reed is 13 feet. The method is: square half the side length of the pond, subtract one foot squared (the part of the reed protruding above water), double the remainder, and divide by two to get the depth of the water. Then add one foot (the part of the reed protruding above water) to get the length of the reed.
〔七〕今有立木，系索其末，委地三尺。引索却行，去本八尺而索尽。问索长几何？
荅曰：一丈二尺、六分尺之一。
术曰：以去本自乘，令如委数而一，所得，加委地数而半之，即索长
现在有一根立着的木柱，将绳子系在它的顶端，并让它自然地垂到地面上三尺。然后，拉直绳子后退，当离开木柱根部八尺时，绳子刚好被拉紧。问绳子有多长？
回答是：一丈二尺又六分之一尺。
解法是：将离开木柱根部的距离自乘，然后用委地数除以这个结果，得到的数字加上委地数后除以二，就是绳子的长度。
There is a standing post, with a rope tied to its top and allowed to drape to the ground for three feet. The rope is then stretched straight back, reaching its limit when eight feet away from the base of the post. How long is the rope? The answer is: one zhang, two chi, and one-sixth of a chi. The method is: square the distance from the base of the post, then divide by the number of feet it is allowed to sag, add this result to the number of feet it is allowed to sag, and halve that sum to find the length of the rope.
〔八〕今有垣高一丈。倚木於垣，上与垣齐。引木却行一尺，其木至地。问木几何？
荅曰：五丈五寸。术曰：以垣高十尺自乘，如却行尺数而一，所得，以加却行尺数而半之，即木长数。
现有一堵墙高十尺，将一根木棒倚靠在墙上，其顶端与墙齐平。如果将木棒后退一尺，那么木棒就会倒在地上。问这根木棒有多长？
答案是：五丈五寸。
方法是：将墙的高度乘以十尺，再除以上退后的距离一尺，所得的结果加上退后的距离的一半，就是木棒的长度。
We have a wall that is 10 feet high. A wooden pole leans against the wall, with its top aligned with the top of the wall. If we move the pole back one foot, it touches the ground. How long is the pole? The answer is: five丈 and five inches (approximately 16.5 feet). The method is: square the height of the wall in feet, divide by the number of feet the pole is moved back, add half of the distance the pole is moved back to this result, and that gives us the length of the pole.
〔九〕今有圆材，埋在壁中，不知大小。以鐻鐻之，深一寸，鐻道长一尺。问径几何？
荅曰：材径二尺六寸。
术曰：半鐻道自乘，如深寸而一，以深寸增之，即材径。
现在有一个圆形的木材，被埋在墙壁中，不知道它的大小。用一个卡尺去测量它，深入墙壁一寸，卡尺的读数是一尺。问这个圆材的直径是多少？
回答是：材料的直径是二尺六寸。
解法是：取卡尺读数的一半进行平方，然后除以深入墙壁的深度（一寸），最后将这个结果加上深入墙壁的深度（一寸），就是材料的直径。
There is a round timber embedded in the wall, and its size is unknown. It is measured with a caliper, penetrating one inch into the wall, and the caliper's reading is one foot. What is the diameter of this round timber? The answer is: The diameter of the material is two feet and six inches. The method is: Square half of the caliper's reading, then divide by the depth of penetration into the wall (one inch), and add this result to the depth of penetration (one inch) to find the diameter of the material.
〔十〕今有开门去閫一尺，不合二寸。问门广几何？
荅曰：一丈一寸。
术曰：以去閫一尺自乘，所得，以不合二寸半之而一，所得，增不合之半，即得门广。
现有一扇门，门框距离地面一尺，两扇门之间不合二寸。问门的宽度是多少？
答案是：一丈一寸。
方法是：将门框距离地面的一尺自乘，所得结果，加上不合的二寸的一半后再除以一，这样得到的结果再加上不合的二寸的一半，就是门的宽度。
We have a door that is one foot away from the threshold, and there is a gap of two inches between the doors. What is the width of the door? The answer is: one zhang and one inch. The method is: square the distance of one foot from the threshold, add half of the two-inch gap and then divide by one, to the result add half of the two-inch gap, and that gives the width of the door.
〔一一〕今有户高多於广六尺八寸，两隅相去適一丈。问户高、广各几何？
荅曰：广二尺八寸；
高九尺六寸。
术曰：令一丈自乘为实。半相多，令自乘，倍之，减实，半其余。以开方除之，所得，减相多之半，即户广。加相多之半，即户高。
现在有一个长方形门，其高度比宽度多六尺八寸，对角线长度恰好是一丈。问这个门的高度和宽度各是多少？
回答是：宽度是二尺八寸；高度是九尺六寸。
解法是：让一丈自乘作为被除数。取对角线差的一半，让它自乘，再加倍，然后减去被除数，把剩下的数开方。用开方后的结果减去对角线差的一半，就是门的宽度。将对角线差的一半加到宽度上，就是门的高度。
There is a rectangular door that is six chi and eight cun taller than its width, and the diagonal length is exactly ten zhang. What are the height and width of the door? The answer is: The width is two chi and eight cun; the height is nine chi and six cun. The method is: Square the ten zhang to get the dividend. Take half of the difference in dimensions, square it, multiply by two, subtract from the dividend, and take the square root of the remainder. Subtracting half of the difference from this gives the width of the door. Adding half of the difference to the width gives the height of the door.
〔一二〕今有户不知高广，竿不知长短。横之不出四尺，从之不出二尺，邪之適出。问户高、广、袤各几何？
荅曰：广六尺，高八尺，袤一丈。术曰：从、横不出相乘，倍，而开方除之。所得加从不出即户广，加横不出即户高，两不出加之，得户袤。
现有一扇门，不知道其高度和宽度，有一根竿子也不知道其长度。横着放竿子，竿子比门宽四尺；竖着放竿子，竿子比门高两尺；斜着放竿子正好可以放在门外。问这扇门的高度、宽度和对角线长度各是多少？
答案是：宽度六尺，高度八尺，对角线长度一丈。
方法是：将竖着和横着放置竿子时多出的长度相乘，再翻倍后开方除以2。得到的结果加上竖着放置时多出的长度即为门的宽度，加上横着放置时多出的长度即为门的高度，两者相加即得到门的对角线长度。
We have a door, and we do not know its height or width. There is a pole, and we do not know its length. Placed horizontally, the pole extends four feet beyond the door; placed vertically, it extends two feet beyond the door; placed diagonally, it fits exactly outside the door. What are the height, width, and diagonal length of the door? The answer is: The width is six feet, the height is eight feet, and the diagonal length is ten feet. The method is: multiply the excess lengths when the pole is placed vertically and horizontally, double the result, and then take the square root and divide by two. Adding the excess length when placed vertically gives the width of the door, adding the excess length when placed horizontally gives the height of the door, and adding both excess lengths gives the diagonal length of the door.
〔一三〕今有竹高一丈，末折抵地，去本三尺。问折者高几何？
荅曰：四尺、二十分尺之十一。术曰：以去本自乘，令如高而一，所得，以减竹高而半其余，即折者之高也。
现在有一根竹竿高一丈，它的末端折下来接触地面，距离竹根三尺。问折断的那一段高度是多少？
回答是：四尺又二十分之一尺。
解法是：将离竹根的距离自乘，然后除以竹竿的高度，用这个结果减去竹竿的总高度，再将剩余的数开方，就是折断部分的高度。
There is a bamboo pole one zhang in height, and when its top is broken and bent to the ground, it is three chi away from the base. What is the height of the broken part? The answer is: four chi and one-twentieth of a chi. The method is: Square the distance from the base, then divide by the height of the bamboo pole. Subtract this result from the total height of the bamboo pole and take the square root of the remainder to find the height of the broken part.
〔一四〕今有二人同所立。甲行率七，乙行率三。乙东行。甲南行十步而邪东北与乙会。问甲乙行各几何？
荅曰：
乙东行一十步半；
甲邪行一十四步半及之。
术曰：令七自乘，三亦自乘，并而半之，以为甲邪行率。邪行率减於七自乘，余为南行率。以三乘七为乙东行率。置南行十步，以甲邪行率乘之，副置十步，以乙东行率乘之，各自为实。实如南行率而一，各得行数。
现有两人在同一起点出发。甲的行进速度是乙的两倍。乙向东行，甲向南方行了十步然后斜向东北方向与乙会合。问甲乙各自行走了多少步？
答案是：
乙向东行了十步半；
甲斜行了十四步半追上了乙。
方法是：将甲的速度七自乘，乙的速度三也自乘，然后将两者相加并除以二，得到甲斜行的速度。用甲斜行的速度减去七的平方，余数就是甲向南行的速度。用三乘以七得到乙向东行的速度。假设甲向南行了十步，用甲斜行的速度乘以十步，同时将十步乘以乙向东行的速度，各自作为实际行进的步数。实际行进的步数除以南行的速度，得到各自的行进步数。
We have two people starting from the same spot. The walking rate of A is seven, and B's rate is three. B walks eastward. After A walks ten steps south, he diagonally heads northeast to meet up with B. How many steps did A and B each walk? The answer is: B walked 10.5 steps eastward; A diagonally walked 14.5 steps to catch up with B. The method is: Square A's rate of seven, square B's rate of three, add them together and halve the result, which will be A's diagonal walking rate. Subtract the square of seven from A's diagonal walking rate, and the remainder is A's southern walking rate. Multiply three by seven to get B's eastern walking rate. Assume that A walked ten steps south, multiply these steps by A's diagonal walking rate, and in parallel, multiply ten steps by B's eastern walking rate, each as their actual steps walked. Divide the actual steps by the southern walking rate, and you will get the number of steps each person walked.
〔一五〕今有句五步，股十二步。问句中容方几何？
荅曰：方三步、十七分步之九。
术曰：并句、股为法，句股相乘为实，实如法而一，得方一步。
现在有一个直角三角形，其中一条直角边长度为五步，另一条直角边长度为十二步。问这个直角三角形的斜边上能容纳多大的正方形？
回答是：三步又十九分之一。
解法是：将两条直角边的长度相加作为除数，两条直角边的长度相乘作为被除数，用被除数除以除数，得到的结果就是斜边上能容纳的正方形的一边的长度。
There is a right-angled triangle with one side being five paces long and the other side twelve paces long. What is the area of the square that can be inscribed on the hypotenuse? The answer is: three paces and one-nineteenth of a pace. The method is: Add the lengths of the two sides to get the divisor, and multiply the lengths of the two sides to get the dividend. Divide the dividend by the divisor to obtain the length of one side of the square inscribed on the hypotenuse.
〔一六〕今有句八步，股十五步。问句中容圆，径几何？
荅曰：六步。
术曰：八步为句，十五步为股，为之求弦。三位并之为法，以句乘股，倍之为实。实如法得径一步。
现有直角三角形，其中勾（一条直角边）为八步，股（另一条直角边）为十五步。问直角三角形的内切圆直径是多少？
答案是：六步。
方法是：将勾的长度作为一条直角边，股的长度作为另一条直角边，求斜边（即弦）。将三边的平方相加得到法则数，用勾的长度乘以股的长度再翻倍得到实数。将实数除以法则数得到内切圆的直径一步长。
We have a right-angled triangle with one leg (the 'gou') being eight steps and the other leg (the 'gu') being fifteen steps. What is the diameter of the inscribed circle (the 'jing')? The answer is: six steps. The method is: Take the length of the gou as one leg and the length of the gu as the other leg to find the hypotenuse (the 'xian'). Add up the squares of all three sides to get the 'fa' (the rule number). Multiply the length of the gou by the length of the gu and double it to get the 'shi' (the actual number). Divide the actual number by the rule number to get the diameter of the inscribed circle, which is one step.
〔一七〕今有邑方二百步，各中开门。出东门十五步有木。问出南门几何步而见木？
荅曰：六百六十六步、太半步。
术曰：出东门步数为法，半邑方自乘为实，实如法得一步。
现在有一个边长为200步的正方形城镇，每个边的中点都有门。从东门出去15步有一棵树。问从南门出去多少步可以看到这棵树？
回答是：六百六十六步又三分之二步。
解法是：以出东门的步数作为除数，将城镇边长的一半自乘作为被除数，用被除数除以除数得到一步。
There is a square town with sides of two hundred steps, and there is a gate in the middle of each side. There is a tree fifteen steps outside the east gate. How many steps from the south gate will the tree be visible? The answer is: six hundred sixty-six and a half steps. The method is: Take the number of steps from the east gate as the divisor, square half the length of the town as the dividend, and divide the dividend by the divisor to get the number of steps.
〔一八〕今有邑，东西七里，南北九里，各中开门。出东门十五里有木。问出南门几何步而见木？
荅曰：三百一十五步。
术曰：东门南至隅步数，以乘南门东至隅步数为实。以木去门步数为法。实如法而一。
有一个城镇，东西宽七里，南北长九里，每边的中点都有门。从东门出去15里有一棵树。问从南门出去多少步可以看到这棵树？
回答是：三百一十五步。
解法是：将南门到东角（东南角）的步数乘以东门到南角（东南角）的步数作为被除数。以树到门的步数作为除数。用被除数除以除数得到一步。
There is a town that is seven li in width from east to west and nine li in length from north to south, with gates at the center of each side. There is a tree 15 li away from the east gate. How many steps from the south gate will the tree be visible? The answer is: three hundred fifteen steps. The method is: Multiply the number of steps from the south gate to the southeast corner by the number of steps from the east gate to the southeast corner to get the dividend. Use the number of steps from the tree to the gate as the divisor. Divide the dividend by the divisor to obtain the number of steps.
〔一九〕今有邑方不知大小，各中开门。出北门三十步有木，出西门七百五十步见木。问邑方几何？
荅曰：一里。
术曰：令两出门步数相乘，因而四之，为实。开方除之，即得邑方。
现有一个正方形的城镇，不知道它的大小，每个方向都开了门。从北门出去三十步有一棵树，从西门出去七百五十步见到同样的树。问这个城镇的边长有多少？
答案是：一里。
方法是：将两个门外步数相乘，然后乘以四，得到实数。对这个实数开方，就可以得到城镇的边长。
We have a square town of unknown size, with gates in the middle of each side. There is a tree 30 steps outside the north gate and a tree 750 steps outside the west gate. What is the side length of the town? The answer is: one li (approximately 0.5 kilometers or 1640 feet). The method is: Multiply the number of steps from both gates, then quadruple it to get the actual number. Take the square root of this number to obtain the side length of the town.
〔二０〕今有邑方不知大小，各中开门。出北门二十步有木。出南门十四步，折而西行一千七百七十五步见木。问邑方几何？
荅曰：二百五十步。
术曰：以出北门步数乘西行步数，倍之，为实。并出南门步数为从法，开方除之，即邑方。
现在有一个正方形城镇，不知道它的大小，每个边的中点都有门。从北门出去20步有一棵树。从南门出去14步，然后转向西走1775步看到了树。问这个城镇的边长是多少？
回答是：250步。
解法是：将出北门的步数乘以向西行的步数，再加倍，作为被除数。加上出南门的步数作为除数，开方后除以这个数，就是这个城镇的边长。
There is a square town of unknown size, with gates in the middle of each side. There is a tree 20 steps away from the north gate. From the south gate, one walks 14 steps and then turns west, walking 1775 steps to see the tree. What is the length of the sides of the town? The answer is: 250 steps. The method is: Multiply the number of steps taken from the north gate by the number of steps walked to the west, double this product to get the dividend. Add the number of steps taken from the south gate to get the divisor, and take the square root of this sum. Dividing the dividend by this number gives the length of the town's sides.
〔二一〕今有邑方十里，各中开门。甲乙俱从邑中央而出。乙东出；甲南出，出门不知步数，邪向东北磨邑，適与乙会。率甲行五，乙行三。问甲、乙行各几何？
荅曰：
甲出南门八百步，邪东北行四千八百八十七步半，及乙。
乙东行四千三百一十二步半。
术曰：令五自乘，三亦自乘，并而半之，为邪行率。邪行率减於五自乘者，余，为南行率。以三乘五，为乙东行率。置邑方半之，以南行率乘之，如东行率而一，即得出南门步数。以增邑方半，即南行。置南行步求弦者，以邪行率乘之，求东者以东行率乘之，各自为实。实如南行率得一步。
现有一个边长为十里的正方形城镇，每面正中都有门。甲和乙都从城镇中心出发。乙向东出城；甲向南出城，出了城门后不知道走了多少步，然后斜向东北方向沿着城墙走，正好与乙相遇。甲的速度是乙的五分之三。问甲和乙各自走了多少步？
答案是：
甲从南门出去走了八百步，然后斜向东北方向走了四千八百八十七步半，追上了乙。
乙向东走了四千三百一十二步半。
方法是：将甲的速度五自乘，乙的速度三也自乘，然后将两者相加并除以二，得到甲斜行的速度。用甲斜行的速度减去五的平方，余数就是甲向南行的速度。用三乘以五得到乙向东行的速度。假设城镇的一半边长，用甲向南行的速度乘以它，再除以乙向东行的速度，就可以得到甲出南门的步数。再加上城镇的一半边长，就是甲向南行的步数。假设甲从南门出发走了一些步数然后斜行，那么用甲斜行的速度乘以这个步数，求得甲追乙的步数；用乙向东行的速度乘以这个步数，得到乙走的步数。实际步数除以南行速度，就可以得到每一步的距离。
We have a square town with sides of ten li, and there are gates in the middle of each side. Both A and B start from the center of the town. B goes out to the east; A goes out to the south, and after an unknown number of steps, A diagonally heads towards the northeast along the town wall and happens to meet B. The ratio of A's speed to B's is five to three. How many steps did A and B each walk? The answer is: A walked 800 steps out of the south gate and then 4887.5 steps diagonally to the northeast to catch up with B. B walked 4312.5 steps to the east. The method is: Square A's speed of five, square B's speed of three, add them together and halve the result to get A's diagonal walking rate. Subtract the square of five from A's diagonal walking rate, and the remainder is A's southern walking rate. Multiply three by five to get B's eastern walking rate. Assume half the side length of the town, multiply it by A's southern walking rate, and divide by B's eastern walking rate to get the number of steps A walked out of the south gate. Add this to half the side length of the town to get the number of steps A walked to the south. Assume that A walked some steps from the south gate and then walked diagonally; multiply these steps by A's diagonal walking rate to find the steps taken to catch up with B; multiply by B's eastern walking rate to find the steps B took. Actual steps divided by the southern walking rate give the distance of one step.
〔二二〕有木去人不知远近。立四表，相去各一丈，令左两表与所望参相直。从后右表望之，入前右表三寸。问木去人几何？
荅曰：三十三丈三尺三寸、少半寸。
术曰：令一丈自乘为实，以三寸为法，实如法而一。
有一棵树，不知道距离人多远。立四个标记，相互之间各隔一丈，让左边的两个标记与所望的树成一条直线。从后面的右边标记望去，进入前面右边标记三寸。问树离人多远？
回答是：三十三丈三尺三寸又一半寸。
解法是：将一丈自乘作为被除数，以三寸作为除数，用被除数除以除数得到结果。
There is a tree whose distance from a person is unknown. Four markers are placed, each one zhang apart, with the left two markers in direct alignment with the tree. When looking from the right marker behind, it enters three inches inside the right marker in front. How far is the tree from the person? The answer is: thirty-three zhang, three chi, and three and a half cun. The method is: Square one zhang to get the dividend, and use three cun as the divisor. Divide the dividend by the divisor to get the result.
〔二三〕有山居木西，不知其高。山去木五十三里，木高九丈五尺。人立木东三里，望木末適与山峰斜平。人目高七尺。问山高几何？
荅曰：一百六十四丈九尺六寸、太半寸。
术曰：置木高减人目高七尺，余，以乘五十三里为实。以人去木三里为法。实如法而一，所得，加木高即山高。
有一座山，山上有棵树向西倾斜，不知道树的高度。山离树53里，树高9丈5尺。人在树东3里的地方站立，望向树梢正好和山峰斜平。人的眼睛高度是7尺。问山的高度是多少？
回答是：164丈9尺6寸又三分之二寸。
解法是：将树的高度减去人眼的高度7尺，余下的数乘以53里作为被除数。以人离树的距离3里作为除数。用被除数除以除数得到的结果加上树的高度就是山的高度。
There is a mountain with a tree leaning to the west, and its height is unknown. The mountain is 53 li away from the tree, which is 9 zhang and 5 chi high. A person stands 3 li east of the tree, looking at the top of the tree just level with the peak of the mountain. The person's eye height is 7 chi. What is the height of the mountain? The answer is: 164 zhang, 9 chi, and 6 cun, plus two-thirds of cun. The method is: Subtract the person's eye height of 7 chi from the height of the tree, multiply the remaining value by 53 li to get the dividend. Use the distance of 3 li from the person to the tree as the divisor. Divide the dividend by the divisor, and add the result to the height of the tree to find the height of the mountain.
〔二四〕今有井径五尺，不知其深。立五尺木於井上，从木末望水岸，入径四寸。问井深几何？
荅曰：五丈七尺五寸。术曰：置井径五尺，以入径四寸减之，余，以乘立木五尺为实。以入径四寸为法。实如法得一寸。
现在有一口井，直径五尺，不知道它的深度。在井上立一根五尺高的木杆，从木杆顶端望向水面的边沿，进入视野的直径是四寸。问井深多少？
回答是：五丈七尺五寸。
解法是：将井的直径五尺，减去进入视野的四寸，余下的数乘以立木的五尺作为被除数。以进入视野的四寸作为除数。用被除数除以除数得到的结果就是井深。
There is a well with a diameter of five feet, and its depth is unknown. A five-foot wooden pole is placed on top of the well, and from the top of the pole, the edge of the water's surface appears to be four inches in diameter. How deep is the well? The answer is: five zhang, seven chi, and five cun. The method is: Subtract the four inches that enter the diameter from the well's diameter of five feet, multiply the remaining value by the height of the wooden pole set up (five feet) to get the dividend. Use the entering diameter of four inches as the divisor. Dividing the dividend by the divisor gives the depth of the well.